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## The Fractional Fourier Transform and Its Applications

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**The Fractional Fourier Transform and Its Applications**Presenter: Pao-Yen Lin Research Advisor: Jian-Jiun Ding , Ph. D. Assistant professor Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Outlines**• Introduction • Fractional Fourier Transform (FrFT) • Linear Canonical Transform (LCT) • Relations to other Transformations • Applications Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Introduction**• Generalization of the Fourier Transform • Categories of Fourier Transform a) Continuous-time aperiodic signal b) Continuous-time periodic signal (FS) c) Discrete-time aperiodic signal (DTFT) d) Discrete-time periodic signal (DFT) Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Fractional Fourier Transform (FrFT)**• Notation • is a transform of • is a transform of Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Fractional Fourier Transform (FrFT) (cont.)**• Constraints of FrFT • Boundary condition • Additive property Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Definition of FrFT**• Eigenvalues and Eigenfunctions of FT • Hermite-Gauss Function Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Definition of FrFT (cont.)**• Eigenvalues and Eigenfunctions of FT Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Definition of FrFT (cont.)**• Eigenvalues and Eigenfunctions of FrFT Use the same eigenfunction but α order eigenvalues Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Definition of FrFT (cont.)**• Kernel of FrFT Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Definition of FrFT (cont.)**Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Properties of FrFT**• Linear. • The first-order transform corresponds to the conventional Fourier transform and the zeroth-order transform means doing no transform. • Additive. Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Linear Canonical Transform (LCT)**• Definition where Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Linear Canonical Transform (LCT) (cont.)**• Properties of LCT • When , the LCT becomes FrFT. • Additive property where Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to other Transformations**• Wigner Distribution • Chirp Transform • Gabor Transform • Gabor-Wigner Transform • Wavelet Transform • Random Process Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Wigner Distribution**• Definition • Property Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Wigner Distribution**Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Wigner Distribution (cont.)**• WD V.S. FrFT • Rotated with angle Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**slope=**Relation to Wigner Distribution (cont.) • Examples Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Chirp Transform**• for Note that is the same as rotated by Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Chirp Transform (cont.)**• Generally, Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Gabor Transform (GT)**• Special case of the Short-Time Fourier Transform (STFT) • Definition Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Gabor Transform (GT) (cont.)**• GT V.S. FrFT • Rotated with angle Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Gabor Transform (GT) (cont.)**• Examples (a)GT of (b)GT of (c)GT of (d)WD of Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**GT V.S. WD**• GT has no cross term problem • GT has less complexity • WD has better resolution • Solution: Gabor-Wigner Transform Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Gabor-Wigner Transform (GWT)**• Combine GT and WD with arbitrary function Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Gabor-Wigner Transform (GWT) (cont.)**• Examples • In (a) • In (b) Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Gabor-Wigner Transform (GWT) (cont.)**• Examples • In (c) • In (d) Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Wavelet Transform**• The kernels of Fractional Fourier Transform corresponding to different values of can be regarded as a wavelet family. Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Random Process**• Classification • Non-Stationary Random Process • Stationary Random Process • Autocorrelation function, PSD are invariant with time t Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Random Process (cont.)**• Auto-correlation function • Power Spectral Density (PSD) Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Random Process (cont.)**• FrFT V.S. Stationary random process • Nearly stationary Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Random Process (cont.)**• FrFT V.S. Stationary random process for Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Random Process (cont.)**• FrFT V.S. Stationary random process PSD: Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Random Process (cont.)**• FrFT V.S. Non-stationary random process Auto-correlation function PSD rotated with angle Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Random Process (cont.)**• Fractional Stationary Random Process If is a non-stationary random process but is stationary and the autocorrelation function of is independent of , then we call the -order fractional stationary random process. Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Relation to Random Process (cont.)**• Properties of fractional stationary random process • After performing the fractional filter, a white noise becomes a fractional stationary random process. • Any non-stationary random process can be expressed as a summation of several fractional stationary random process. Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Applications of FrFT**• Filter design • Optical systems • Convolution • Multiplexing • Generalization of sampling theorem Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**noise**noise signal Filter design using FrFT • Filtering a known noise • Filtering in fractional domain Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Filter design using FrFT (cont.)**• Random noise removal If is a white noise whose autocorrelation function and PSD are: After doing FrFT Remain unchanged after doing FrFT! Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**signal**signal Filter design using FrFT (cont.) • Random noise removal • Area of WD ≡ Total energy Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Optical systems**• Using FrFT/LCT to Represent Optical Components • Using FrFT/LCT to Represent the Optical Systems • Implementing FrFT/LCT by Optical Systems Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Using FrFT/LCT to Represent Optical Components**• Propagation through the cylinder lens with focus length • Propagation through the free space (Fresnel Transform) with length Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**input**output Using FrFT/LCT to Represent the Optical Systems Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Implementing FrFT/LCT by Optical Systems**• All the Linear Canonical Transform can be decomposed as the combination of the chirp multiplication and chirp convolution and we can decompose the parameter matrix into the following form Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**input**output input output Implementing FrFT/LCT by Optical Systems (cont.) The implementation of LCT with 2 cylinder lenses and 1 free space The implementation of LCT with 1 cylinder lens and 2 free spaces Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Convolution**• Convolution in domain • Multiplication in domain Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Convolution (cont.)**Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Multiplexing using FrFT**TDM FDM Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Multiplexing using FrFT**Inefficient multiplexing Efficient multiplexing Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University**Generalization of sampling theorem**• If is band-limited in some transformed domain of LCT, i.e., then we can sample by the interval as Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University